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Quadrupedal running with a flexible torso: control and speed transitions with sums-of-squares verification

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Abstract

This paper studies quadrupedal bounding in the presence of flexible torso and compliant legs with non-trivial inertia, and it proposes a method for speed transitions by sequentially composing locally stable bounding motions corresponding to different speeds. First, periodic bounding motions are generated simply by positioning the legs during flight via suitable (virtual) holonomic constraints imposed on the leg angles; at this stage, no control effort is developed on the support legs, producing efficient, nearly passive, bounding gaits. The resulting motions are then stabilized by a hybrid control law which coordinates the movement of the torso and legs in continuous time, and updates the leg touchdown angles in an event-based fashion. Finally, through sums-of-squares programming, formally verified estimates of the domain of attraction of stable fixed points are employed to realize stable speed transitions by switching among different bounding gaits in a sequential fashion.

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Notes

  1. Given the relatively high stiffness value of the leg spring, this assumption represents a reasonable approximation of the liftoff condition, which occurs when the vertical component of the ground reaction force becomes zero and the vertical acceleration of the toe is directed upwards.

  2. Note that \(\mathcal{D}_0\) and \(\mathcal{D}_1\) do not represent the domains of attraction of the entire periodic orbits \({ \phi }_0\) and \({ \phi }_1\).

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Delaware, Newark, DE, USA

    Qu Cao & Ioannis Poulakakis

Authors
  1. Qu Cao
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  2. Ioannis Poulakakis
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Corresponding author

Correspondence to Ioannis Poulakakis.

Additional information

This work is supported in part by NSF CAREER Award IIS-135072 and ARL contract W911NF-12-1-0117.

This work was presented in part at the 1st International Symposium on Swarm Behavior and Bio-Inspired Robotics, Kyoto, Japan, October 28–30, 2015.

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Cao, Q., Poulakakis, I. Quadrupedal running with a flexible torso: control and speed transitions with sums-of-squares verification. Artif Life Robotics 21, 384–392 (2016). https://doi.org/10.1007/s10015-016-0330-5

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  • DOI: https://doi.org/10.1007/s10015-016-0330-5

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